Integrand size = 27, antiderivative size = 266 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {a x}{b^3}+\frac {2 a \left (2 a^2-3 b^2\right ) x}{b^5}+\frac {2 \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 d}-\frac {2 \left (a^2-b^2\right )^{3/2} \left (5 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 b^5 d}-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {\cos (c+d x)}{b^2 d}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{b^4 d}-\frac {\cos ^3(c+d x)}{3 b^2 d}-\frac {a \cos (c+d x) \sin (c+d x)}{b^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a b^4 d (a+b \sin (c+d x))} \] Output:
a*x/b^3+2*a*(2*a^2-3*b^2)*x/b^5+2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*d*x+ 1/2*c))/(a^2-b^2)^(1/2))/b^5/d-2*(a^2-b^2)^(3/2)*(5*a^2+b^2)*arctan((b+a*t an(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^2/b^5/d-arctanh(cos(d*x+c))/a^2/d+co s(d*x+c)/b^2/d+3*(a^2-b^2)*cos(d*x+c)/b^4/d-1/3*cos(d*x+c)^3/b^2/d-a*cos(d *x+c)*sin(d*x+c)/b^3/d+(a^2-b^2)^2*cos(d*x+c)/a/b^4/d/(a+b*sin(d*x+c))
Time = 2.23 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {12 a \left (4 a^2-5 b^2\right ) (c+d x)}{b^5}-\frac {24 \left (a^2-b^2\right )^{3/2} \left (4 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 b^5}+\frac {9 \left (4 a^2-3 b^2\right ) \cos (c+d x)}{b^4}-\frac {\cos (3 (c+d x))}{b^2}-\frac {12 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a^2}+\frac {12 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^2}+\frac {12 \left (a^2-b^2\right )^2 \cos (c+d x)}{a b^4 (a+b \sin (c+d x))}-\frac {6 a \sin (2 (c+d x))}{b^3}}{12 d} \] Input:
Integrate[(Cos[c + d*x]^5*Cot[c + d*x])/(a + b*Sin[c + d*x])^2,x]
Output:
((12*a*(4*a^2 - 5*b^2)*(c + d*x))/b^5 - (24*(a^2 - b^2)^(3/2)*(4*a^2 + b^2 )*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*b^5) + (9*(4*a^2 - 3*b^2)*Cos[c + d*x])/b^4 - Cos[3*(c + d*x)]/b^2 - (12*Log[Cos[(c + d*x)/ 2]])/a^2 + (12*Log[Sin[(c + d*x)/2]])/a^2 + (12*(a^2 - b^2)^2*Cos[c + d*x] )/(a*b^4*(a + b*Sin[c + d*x])) - (6*a*Sin[2*(c + d*x)])/b^3)/(12*d)
Time = 0.68 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3042, 3376, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x) (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3376 |
| \(\displaystyle \int \left (-\frac {2 \left (3 a b^2-2 a^3\right )}{b^5}+\frac {\left (a^2-b^2\right )^3}{a b^5 (a+b \sin (c+d x))^2}-\frac {\left (5 a^2+b^2\right ) \left (a^2-b^2\right )^2}{a^2 b^5 (a+b \sin (c+d x))}+\frac {3 \left (b^2-a^2\right ) \sin (c+d x)}{b^4}+\frac {\csc (c+d x)}{a^2}+\frac {2 a \sin ^2(c+d x)}{b^3}-\frac {\sin ^3(c+d x)}{b^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^5 d}-\frac {2 \left (a^2-b^2\right )^{3/2} \left (5 a^2+b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^2 b^5 d}-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {2 a x \left (2 a^2-3 b^2\right )}{b^5}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{b^4 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a b^4 d (a+b \sin (c+d x))}-\frac {a \sin (c+d x) \cos (c+d x)}{b^3 d}+\frac {a x}{b^3}-\frac {\cos ^3(c+d x)}{3 b^2 d}+\frac {\cos (c+d x)}{b^2 d}\) |
Input:
Int[(Cos[c + d*x]^5*Cot[c + d*x])/(a + b*Sin[c + d*x])^2,x]
Output:
(a*x)/b^3 + (2*a*(2*a^2 - 3*b^2)*x)/b^5 + (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^5*d) - (2*(a^2 - b^2)^(3/2)*(5*a ^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*b^5*d) - ArcTanh[Cos[c + d*x]]/(a^2*d) + Cos[c + d*x]/(b^2*d) + (3*(a^2 - b^2)*Cos[ c + d*x])/(b^4*d) - Cos[c + d*x]^3/(3*b^2*d) - (a*Cos[c + d*x]*Sin[c + d*x ])/(b^3*d) + ((a^2 - b^2)^2*Cos[c + d*x])/(a*b^4*d*(a + b*Sin[c + d*x]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(d*sin[ e + f*x])^n*(a + b*sin[e + f*x])^m*(1 - sin[e + f*x]^2)^(p/2), x], x] /; Fr eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && ( LtQ[m, -1] || (EqQ[m, -1] && GtQ[p, 0]))
Time = 11.42 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.18
| method | result | size |
| derivativedivides | \(\frac {\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}+\frac {\frac {4 \left (\frac {a \,b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2}+\left (\frac {3}{2} a^{2} b -\frac {3}{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (3 a^{2} b -2 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {a \,b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {3 a^{2} b}{2}-\frac {7 b^{3}}{6}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+2 a \left (4 a^{2}-5 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}-\frac {4 \left (\frac {-\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{5} b}{2}+a^{3} b^{3}-\frac {a \,b^{5}}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (4 a^{6}-7 a^{4} b^{2}+2 a^{2} b^{4}+b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{2} b^{5}}}{d}\) | \(314\) |
| default | \(\frac {\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}+\frac {\frac {4 \left (\frac {a \,b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2}+\left (\frac {3}{2} a^{2} b -\frac {3}{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (3 a^{2} b -2 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {a \,b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {3 a^{2} b}{2}-\frac {7 b^{3}}{6}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+2 a \left (4 a^{2}-5 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}-\frac {4 \left (\frac {-\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{5} b}{2}+a^{3} b^{3}-\frac {a \,b^{5}}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (4 a^{6}-7 a^{4} b^{2}+2 a^{2} b^{4}+b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{2} b^{5}}}{d}\) | \(314\) |
| risch | \(\frac {4 a^{3} x}{b^{5}}-\frac {5 a x}{b^{3}}+\frac {3 i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{3}}+\frac {3 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{2 d \,b^{4}}-\frac {9 \,{\mathrm e}^{i \left (d x +c \right )}}{8 b^{2} d}+\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{2 d \,b^{4}}-\frac {9 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 b^{2} d}+\frac {i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{4 b^{3} d}-\frac {4 i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right ) a^{2}}{d \,b^{5}}-\frac {i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{4 b^{3} d}+\frac {2 i \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left (-i a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b^{5} d a \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )}-\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d b \,a^{2}}+\frac {4 i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right ) a^{2}}{d \,b^{5}}-\frac {3 i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{3}}+\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d b \,a^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{2}}-\frac {\cos \left (3 d x +3 c \right )}{12 d \,b^{2}}\) | \(576\) |
Input:
int(cos(d*x+c)^5*cot(d*x+c)/(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/a^2*ln(tan(1/2*d*x+1/2*c))+4/b^5*((1/2*a*b^2*tan(1/2*d*x+1/2*c)^5+( 3/2*a^2*b-3/2*b^3)*tan(1/2*d*x+1/2*c)^4+(3*a^2*b-2*b^3)*tan(1/2*d*x+1/2*c) ^2-1/2*a*b^2*tan(1/2*d*x+1/2*c)+3/2*a^2*b-7/6*b^3)/(1+tan(1/2*d*x+1/2*c)^2 )^3+1/2*a*(4*a^2-5*b^2)*arctan(tan(1/2*d*x+1/2*c)))-4/a^2/b^5*((-1/2*b^2*( a^4-2*a^2*b^2+b^4)*tan(1/2*d*x+1/2*c)-1/2*a^5*b+a^3*b^3-1/2*a*b^5)/(tan(1/ 2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)+1/2*(4*a^6-7*a^4*b^2+2*a^2*b^4+ b^6)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/ 2))))
Time = 0.39 (sec) , antiderivative size = 698, normalized size of antiderivative = 2.62 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^5*cot(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
[1/6*(4*a^3*b^3*cos(d*x + c)^3 + 6*(4*a^6 - 5*a^4*b^2)*d*x - 3*(4*a^5 - 3* a^3*b^2 - a*b^4 + (4*a^4*b - 3*a^2*b^3 - b^5)*sin(d*x + c))*sqrt(-a^2 + b^ 2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2 *(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos (d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) + 6*(4*a^5*b - 5*a^3*b^3 + a*b^5)*cos(d*x + c) - 3*(b^6*sin(d*x + c) + a*b^5)*log(1/2*cos(d*x + c) + 1/2) + 3*(b^6*sin(d*x + c) + a*b^5)*log(-1/2*cos(d*x + c) + 1/2) - 2*(a^2* b^4*cos(d*x + c)^3 - 3*(4*a^5*b - 5*a^3*b^3)*d*x - 6*(a^4*b^2 - a^2*b^4)*c os(d*x + c))*sin(d*x + c))/(a^2*b^6*d*sin(d*x + c) + a^3*b^5*d), 1/6*(4*a^ 3*b^3*cos(d*x + c)^3 + 6*(4*a^6 - 5*a^4*b^2)*d*x + 6*(4*a^5 - 3*a^3*b^2 - a*b^4 + (4*a^4*b - 3*a^2*b^3 - b^5)*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(- (a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) + 6*(4*a^5*b - 5*a^3* b^3 + a*b^5)*cos(d*x + c) - 3*(b^6*sin(d*x + c) + a*b^5)*log(1/2*cos(d*x + c) + 1/2) + 3*(b^6*sin(d*x + c) + a*b^5)*log(-1/2*cos(d*x + c) + 1/2) - 2 *(a^2*b^4*cos(d*x + c)^3 - 3*(4*a^5*b - 5*a^3*b^3)*d*x - 6*(a^4*b^2 - a^2* b^4)*cos(d*x + c))*sin(d*x + c))/(a^2*b^6*d*sin(d*x + c) + a^3*b^5*d)]
Timed out. \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**5*cot(d*x+c)/(a+b*sin(d*x+c))**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^5*cot(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="maxima" )
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.21 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.33 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} + \frac {3 \, {\left (4 \, a^{3} - 5 \, a b^{2}\right )} {\left (d x + c\right )}}{b^{5}} - \frac {6 \, {\left (4 \, a^{6} - 7 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{2} b^{5}} + \frac {2 \, {\left (3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 9 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, a^{2} - 7 \, b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} b^{4}} + \frac {6 \, {\left (a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a^{2} b^{4}}}{3 \, d} \] Input:
integrate(cos(d*x+c)^5*cot(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="giac")
Output:
1/3*(3*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 + 3*(4*a^3 - 5*a*b^2)*(d*x + c)/ b^5 - 6*(4*a^6 - 7*a^4*b^2 + 2*a^2*b^4 + b^6)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt (a^2 - b^2)*a^2*b^5) + 2*(3*a*b*tan(1/2*d*x + 1/2*c)^5 + 9*a^2*tan(1/2*d*x + 1/2*c)^4 - 9*b^2*tan(1/2*d*x + 1/2*c)^4 + 18*a^2*tan(1/2*d*x + 1/2*c)^2 - 12*b^2*tan(1/2*d*x + 1/2*c)^2 - 3*a*b*tan(1/2*d*x + 1/2*c) + 9*a^2 - 7* b^2)/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*b^4) + 6*(a^4*b*tan(1/2*d*x + 1/2*c) - 2*a^2*b^3*tan(1/2*d*x + 1/2*c) + b^5*tan(1/2*d*x + 1/2*c) + a^5 - 2*a^3* b^2 + a*b^4)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)*a^ 2*b^4))/d
Time = 23.64 (sec) , antiderivative size = 5197, normalized size of antiderivative = 19.54 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \] Input:
int((cos(c + d*x)^5*cot(c + d*x))/(a + b*sin(c + d*x))^2,x)
Output:
log(tan(c/2 + (d*x)/2))/(a^2*d) + ((2*(12*a^4 + 3*b^4 - 13*a^2*b^2))/(3*a* b^4) + (2*tan(c/2 + (d*x)/2)*(18*a^4 + 3*b^4 - 20*a^2*b^2))/(3*a^2*b^3) + (2*tan(c/2 + (d*x)/2)^7*(2*a^4 + b^4 - 2*a^2*b^2))/(a^2*b^3) + (2*tan(c/2 + (d*x)/2)^6*(4*a^4 + b^4 - 3*a^2*b^2))/(a*b^4) + (2*tan(c/2 + (d*x)/2)^5* (10*a^4 + 3*b^4 - 12*a^2*b^2))/(a^2*b^3) + (2*tan(c/2 + (d*x)/2)^4*(12*a^4 + 3*b^4 - 13*a^2*b^2))/(a*b^4) + (2*tan(c/2 + (d*x)/2)^3*(14*a^4 + 3*b^4 - 14*a^2*b^2))/(a^2*b^3) + (2*tan(c/2 + (d*x)/2)^2*(36*a^4 + 9*b^4 - 43*a^ 2*b^2))/(3*a*b^4))/(d*(a + 2*b*tan(c/2 + (d*x)/2) + 4*a*tan(c/2 + (d*x)/2) ^2 + 6*a*tan(c/2 + (d*x)/2)^4 + 4*a*tan(c/2 + (d*x)/2)^6 + a*tan(c/2 + (d* x)/2)^8 + 6*b*tan(c/2 + (d*x)/2)^3 + 6*b*tan(c/2 + (d*x)/2)^5 + 2*b*tan(c/ 2 + (d*x)/2)^7)) + (2*a*atan(((a*(4*a^2 - 5*b^2)*((a*(4*a^2 - 5*b^2)*((32* (4*b^16 + 7*a^2*b^14 - 25*a^4*b^12 + 114*a^6*b^10 - 235*a^8*b^8 + 184*a^10 *b^6 - 48*a^12*b^4))/(a^2*b^11) + (32*tan(c/2 + (d*x)/2)*(20*a^4*b^18 - 19 *a^2*b^20 + 610*a^6*b^16 - 1596*a^8*b^14 + 1434*a^10*b^12 - 480*a^12*b^10 + 32*a^14*b^8))/(a^3*b^16) + (a*(4*a^2 - 5*b^2)*((32*(8*a^2*b^16 + 4*a^4*b ^14 - 25*a^6*b^12 + 14*a^8*b^10))/(a^2*b^11) + (a*((32*(4*a^4*b^16 - 3*a^6 *b^14))/(a^2*b^11) + (32*tan(c/2 + (d*x)/2)*(16*a^4*b^22 - 17*a^6*b^20 + 2 *a^8*b^18))/(a^3*b^16))*(4*a^2 - 5*b^2)*1i)/b^5 + (32*tan(c/2 + (d*x)/2)*( 16*a^2*b^22 + 3*a^4*b^20 - 62*a^6*b^18 + 60*a^8*b^16 - 16*a^10*b^14))/(a^3 *b^16))*1i)/b^5)*1i)/b^5 - (32*(5*a^2*b^12 - 224*a^14 - 184*a^4*b^10 + ...
Time = 98.44 (sec) , antiderivative size = 549, normalized size of antiderivative = 2.06 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {-24 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (d x +c \right ) a^{4} b +18 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (d x +c \right ) a^{2} b^{3}+6 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (d x +c \right ) b^{5}-24 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) a^{5}+18 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) a^{3} b^{2}+6 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) a \,b^{4}+\cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b^{4}-2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3} b^{3}+6 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4} b^{2}-7 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{4}+12 \cos \left (d x +c \right ) a^{5} b -13 \cos \left (d x +c \right ) a^{3} b^{3}+3 \cos \left (d x +c \right ) a \,b^{5}+3 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right ) b^{6}+3 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{5}+12 \sin \left (d x +c \right ) a^{5} b d x +6 \sin \left (d x +c \right ) a^{4} b^{2}-15 \sin \left (d x +c \right ) a^{3} b^{3} d x -7 \sin \left (d x +c \right ) a^{2} b^{4}+12 a^{6} d x +6 a^{5} b -15 a^{4} b^{2} d x -7 a^{3} b^{3}}{3 a^{2} b^{5} d \left (\sin \left (d x +c \right ) b +a \right )} \] Input:
int(cos(d*x+c)^5*cot(d*x+c)/(a+b*sin(d*x+c))^2,x)
Output:
( - 24*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))* sin(c + d*x)*a**4*b + 18*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/s qrt(a**2 - b**2))*sin(c + d*x)*a**2*b**3 + 6*sqrt(a**2 - b**2)*atan((tan(( c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)*b**5 - 24*sqrt(a**2 - b **2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a**5 + 18*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a**3*b**2 + 6*sqr t(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a*b**4 + c os(c + d*x)*sin(c + d*x)**3*a**2*b**4 - 2*cos(c + d*x)*sin(c + d*x)**2*a** 3*b**3 + 6*cos(c + d*x)*sin(c + d*x)*a**4*b**2 - 7*cos(c + d*x)*sin(c + d* x)*a**2*b**4 + 12*cos(c + d*x)*a**5*b - 13*cos(c + d*x)*a**3*b**3 + 3*cos( c + d*x)*a*b**5 + 3*log(tan((c + d*x)/2))*sin(c + d*x)*b**6 + 3*log(tan((c + d*x)/2))*a*b**5 + 12*sin(c + d*x)*a**5*b*d*x + 6*sin(c + d*x)*a**4*b**2 - 15*sin(c + d*x)*a**3*b**3*d*x - 7*sin(c + d*x)*a**2*b**4 + 12*a**6*d*x + 6*a**5*b - 15*a**4*b**2*d*x - 7*a**3*b**3)/(3*a**2*b**5*d*(sin(c + d*x)* b + a))